Partial symmetry and existence of least energy solutions to some nonlinear elliptic equations on Riemannian models

TL;DR

This paper investigates the existence and partial symmetry of least energy solutions to nonlinear elliptic equations on Riemannian models, including hyperbolic space, with broad nonlinearities.

Contribution

It establishes partial symmetry and existence results for least energy solutions on general Riemannian models with unbounded sectional geometry.

Findings

Existence of least energy solutions on various Riemannian models.

Partial symmetry properties of solutions.

Applicability to a wide class of nonlinearities.

Abstract

We consider least energy solutions to the nonlinear equation $-\Delta_g u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$ which include the classical hyperbolic space $\mathbb H^n$ as well as manifolds with unbounded sectional geometry. Partial symmetry and existence of least energy solutions is proved for quite general nonlinearities $f(r,u)$, where $r$ denotes the geodesic distance from the pole of $M$.